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Distinct Trapezoidal Decompositions? (Posted on 2025-02-03) Difficulty: 3 of 5
How many distinct trapezoidal decompositions (sum of consecutive positive integers) does n have?

For example, 15 = 7+8 = 4+5+6 = 1+2+3+4+5 has three.

No Solution Yet Submitted by K Sengupta    
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Solution Solution | Comment 3 of 4 |
Let F be the first and term in some trapezoidal decomposition.  Let there be D terms.  Then a formula for the sum of an arithmetic sequence with a common difference of 1 is N = (D/2)*(2F+D-1).

Multiply through by 2 and get 2N = D * (2F+D-1).  Note that the two terms of the product are of opposite parity, therefore at least one is odd.

Then there is a solution whenever 2N has a factorization with an odd number.  But the presence of factors of 2 in 2N has no bearing on any odd factor, so we can refine that to say there is a trapezoidal decomposition for each odd factor of N.
This also includes the trivial decomposition of a single term N=N (when D=1), so we need to subtract 1 from that total.
Thus the number of distinct trapezoidal decompositions of N is one less than the number of odd factors of N.


  Posted by Brian Smith on 2025-02-03 13:15:13
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